Invariants in the cohomology of the complement of quaternionic reflection arrangements
Lorenzo Giordani, Gerhard Roehrle, Johannes Schmitt
TL;DR
The article extends the study of G-actions on the rational cohomology of the complement M(A) of reflection arrangements to quaternionic reflection groups, generalizing the invariant theory of complex reflection groups. It develops an inductive framework based on Brieskorn–Orlik–Solomon theory, the parabolic-subgroup lattice (isomorphic to the Dowling lattice for imprimitive cases), and the Euler-type identities to compute Poincaré polynomials P(A,G; t) for the G-invariant cohomology. A key finding is that, apart from a single new family arising in a particular imprimitive irreducible class, the quaternionic case mirrors the complex case in terms of P(A,G; t) types, despite substantial structural differences at the group level. The paper also provides explicit bases for H^*(M(A))^G in both imprimitive and primitive quaternionic reflection groups, aided by computational tools, and situates these results within broader observations about factorization patterns of cohomology polynomials in the quaternionic setting. These results yield concrete, computable invariants with potential implications for representation theory and topological combinatorics of quaternionic reflection arrangements.
Abstract
Let $\mathcal A$ be a hyperplane arrangement in a vector space $V$ and $G \leq GL(V)$ a group fixing $\mathcal A$. In case when $G$ is a complex reflection group and $\mathcal A=\mathcal A(G)$ is its reflection arrangement in $V$, Douglass, Pfeiffer, and Röhrle studied the invariants of the $Q G$-module $H^*(M(\mathcal A);Q)$, the rational, singular cohomology of the complement space $M(\mathcal A)$ in $V$. In this paper we generalize the work in Douglass, Pfeiffer, and Röhrle to the case of quaternionic reflection groups. We obtain a straightforward generalization of the Hilbert--Poincaré series of the ring of invariants in the cohomology from the complex case when the quaternionic reflection group is complex-reducible according to Cohen's classification. Surprisingly, only one additional family of new types of Poincaré polynomials occurs in the quaternionic setting which is not realised in the complex case, namely those of a particular class of imprimitive irreducible quaternionic reflection groups. Finally, we discuss bases of the space of $G$-invariants in $H^*(M(\mathcal A);Q)$.